Question

Differentiate the following functions w.r.t.x: $fraction numerator cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 2 straight x plus 7 end root end fraction comma space minus 2 less than straight x less than 2$

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Solution

Let space space space space space straight y equals fraction numerator cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 2 straight x plus 7 end root end fraction therefore space dy over dx equals fraction numerator square root of 2 straight x plus 7 end root begin display style straight d over dx end style open parentheses cos to the power of negative 1 end exponent straight x over 2 close parentheses minus cos to the power of negative 1 end exponent straight x over 2 begin display style straight d over dx end style open parentheses square root of 2 straight x plus 7 end root close parentheses over denominator left parenthesis square root of 2 straight x plus 7 end root right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator square root of 2 straight x plus 7 end root. begin display style fraction numerator negative 1 over denominator square root of 1 minus begin display style straight x squared over 4 end style end root end fraction end style. begin display style 1 half end style minus cos to the power of negative 1 end exponent straight x over 2. begin display style fraction numerator 2 over denominator 2 square root of 2 straight x plus 7 end root end fraction end style over denominator 2 straight x plus 7 end fraction space space space space space space space space space space space space space equals fraction numerator negative begin display style fraction numerator square root of 2 straight x plus 7 end root over denominator square root of 4 minus straight x squared end root end fraction end style minus begin display style fraction numerator cos to the power of negative 1 end exponent straight x over 2 over denominator square root of 2 straight x plus 7 end root end fraction end style over denominator 2 straight x plus 7 end fraction equals fraction numerator negative left parenthesis 2 straight x plus 7 right parenthesis minus square root of 4 minus straight x squared end root space cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 4 minus straight x squared end root left parenthesis 2 straight x plus 7 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

therefore space dy over dx equals fraction numerator negative left parenthesis 2 straight x plus 7 right parenthesis minus square root of 4 minus straight x squared end root space cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 4 minus straight x squared end root left parenthesis 2 straight x plus 7 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

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Continuity And Differentiability

Differentiate the following functions w.r.t.x: $fraction numerator cos to the power of negative 1 end exponent begin display style straight x over 2 end style over denominator square root of 2 straight x plus 7 end root end fraction comma space minus 2 less than straight x less than 2$

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